I've been having trouble solving this problem for a long time. i used the variables: a=1 , b = 2 height = x^2 radius : 1+ x and took the integral of it using the cylindrical shell method.

The radius should be  "x - 1" .

The region is revolving around the line x= 1. The radius is distance from this line.  So radius varies from 0 to 1 as x goes from 1 to 2. 
Therefore ,  r = x - 1.

Everything else is correct.

Here is the integral solution:

50.2450.24 in squared